The Completeness Criterion for Closure Operator with the Equality Predicate Branching on the Set of Multioperations on Two-Element Set Текст научной статьи по специальности «Математика»

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

2019. Т. 29. С. 68-85

УДК 519.716 MSG 03В50, 08А99

DOI https://doi.org/10.26516/1997-7670.2019.29.68

The Completeness Criterion for Closure Operator with the Equality Predicate Branching on the Set of Multioperations on Two-Element Set

V. I. Panteleev

Irkutsk State University, Irkutsk, Russian Federation

L. V. Riabets

Irkutsk State University, Irkutsk, Russian Federation

Abstract. Multioperations are operations from a finite set A to set of all subsets of A. The usual composition operator leads to a continuum of closed sets. Therefore, the research of closure operators, which contain composition and other operations becomes necessary. In the paper, the closure of multioperations that can be obtained using the operations of adding dummy variables, identifying variables, composition operator, and operator with the equality predicate branching is studied. We obtain eleven precomplete closed classes of multioperations of rank 2 and prove the completeness criterion. The diagram of inclusions for one of the precomplete class is presented.

Keywords: closure, equality predicate, multioperation, closed set, composition, completeness criterion.

Discrete functions defined on a finite set. A and taking values in the set of subsets of A are widely considered as a generalization of classical functional systems of fc-valued logic. Partial Boolean functions, hyperfunct.ions, and multifunctions with respect to the composition operator were studied in [26; 12; 13; 15].

The superposition operator leads, as a rule, to a countable or continuous classification; therefore, closure operators that generate finite classifications of functions are of interest. Such operators, in particular, include

1. Introduction

the parametric and positive closure operators [7], the operator with the equality predicate branching (_E-closure operator) [8]. An investigation of the last operator on the set of Boolean functions, partial Boolean functions and on the set of functions of fc-valued logic can be found in [8-10]. All ^-closed classes for the set of partial Boolean functions were obtained in [11]. The completeness criterion for the -Enclosing operator on the set of hyperfunctions of rank two was proved in [14].

Suppose E2 = {0,1} and on € E2, i € {1,..., n}; then (a\, ct2, • • •, an) is called a binary set or just a set and is denoted by a. Let n be a length of a. If the length of the binary set a is not indicated, it is determined by context.

By M2 denote a set of all rank two multioperations, and it defines as follows

M2,n = {/ [ / : E'2 -»• 2E2} , M2 = \jM2,n.

n

In what follows, we will not distinguish between a set of one element and an element of this set. For the set E2, we will use the notation "—" (dash) and empty set we will denote as "*". Instead of term the "multioperation", sometimes, we will use the word "operation" if this does not confuse.

The set M2 contains the set of hyperfunctions (#2), the set of partial Boolean functions (O2), and all Boolean functions (O2):

H2,n = {/!/:££->• 2®2 \ {0}} , H2 = \jH2,n,

n

01,n = {f\f:E^E2U {0}} , O^ = \JOln,

n

02,n = {/ I / : E2 —>■ E2} , 02=U02,n.

n

The n-variable multioperation / will be represented as a vector (75,..., tj), where r^ equals to /(<r). Such vectors have the form (/(0) /(1)) and (/(0,0) /(0,1) /(1,0) /(1,1)) for unary and binary multioperations respectively.

Suppose ¡{xi,...,xn), fi(xi,...,xm),...,fn(xi,...,xm) are multioperations.

The g(x 1,..., xm) is said to be S'tz-composition of

/ (/l(^l) • • • ) xm) j • • • j fn(%l j • • • j

if

g(a1,...,am) = (J /(/?!,...,/?„), where (a\,..., am) € E™.

The composition operator defined in that way allows us to find multioperation values on the subsets of 2®2. Moreover, we consider the element of such sets as a constant function.

Example 1. /(0, -, 1) = /(0,0,1) U /(0,1,1) and /(0, *, 1) = *.

We say that the multioperation g(xi,..., xn) is obtained from the functions fi(xi,..., xn), /2(^1, •••, xn) using the operator with the equality predicate branching (_E-operator) if for some i, j € {1,... ,11} the following relation holds:

_ I f\(x\, . . . , Xn), if Xi = Xj] yyx 1,..., Xn) — <,

I j2{Xi,..., xn), otherwise.

The set of all multioperations that can be obtained from the set Q C M2 using the operations of adding dummy variables, identifying variables, Su-composition and I?-operator is called ESu-closure of set Q and is denoted by [Q\.

The function obtained by adding dummy variables will be denoted by the same symbol as the original one. Thus, if g(x) is some multioperation, then g(x, y) is the operation obtained by adding the dummy variable y. In the future, we will not discuss this separately

A set of multioperations that coincides with its closure is called an ESjj-closed class. We say that the set P C Q generates an ESu-closed class Q if [P] = Q. Therefore P is -ESf/-complete in Q.

The P C M2 is said to be the precomplete set in M2 if [P] C M2, but [P U /] = M2 for any f <£ P.

Let Rm be an m-place predicate on 2®2 of the form

Rm = {(an, • • •, aim), (a2i,..., a2m), ■ ■ ■ (ap 1,..., apm)} .

The multioperation f(x 1,..., xn) preserves predicate Rm if for any n sets

Tii} i ■ ■ ■ 1 (,Pnli ■ ■ ■ 1 Pnm)

from the predicate Rm, the set

(/(/? 11, • • • ,Plm), ■■■ , f(Pnl, ■ ■ ■ ,Pnm))

belongs to Rm.

Pol R denotes the set of multioperations preserving R. Moreover, replace predicate containing n sets will be set by a m x n matrix, in which the columns are sets from the predicate.

In the general case, the set of multioperations preserving a particular predicate is not necessarily closed with respect to composition. But the following lemma holds.

Lemma 1. If the operation f is obtained by a composition of operations g, g\, ..., gm that preserve some predicate R, then operation f on binary sets from R will necessarily return a set (not necessarily binary) from the predicate.

2. ESu-closed classes

Consider the following 11 sets of multioperations: K1 = {/|/(0,...,0)e{0,-}};K2 = {/|/(l,...,l)e{l,-}}; K3 = {f I /(0,..., 0) <E {0, *}}; K4 = {f\ /(1,..., 1) <E {1, *}}; K5 = 0*2; K6 = H2- K7 = {f | f(a) € {*, 1, -}};

Ks = {f\ /(a) € {*, 0, -}}; Kg = PolR9] R9 = ( J J * ^

(01** * * 0 1 — 1 0 0 1 - * * * * Ku = {f\ *e/(0,...,0)u/(l,...,l) or /(0,..., 0) =0 and /(1,..., 1) = 1}.

Theorem 1. The sets Ki — Kn are ESu-closed.

Proof. It is easily proved that the sets K\ — Kg are ESu--closed.

Consider the set Kg. Let multioperations f,fi,...,fm preserve Rg and multioperation

g(xI, ... , Xm) = /(/l(xi, . . . , Xn), . . . , fm(% 1) • • • ) %n))

does not preserve the predicate Rg. Then

/0 1 - \ f 0 1 * * *--01-0 ll,

^10-Jn(oi01- 0 1 ** * - - J*0-

Therefore

/0l\(01***--01-0 1 1 .

Hi oJn\o 1 0 1 - 0 1 * * * - -

But this contradicts Lemma 1.

Consider the operator of equality predicate branching for Kg. Let

\ _ I / l(xl} • • • J %m), if Xi = Xj; gy.L 1, . . . , ,Lm) — <,

[j2{xi, ■ ■ ■ ,xm), otherwise.

Suppose that g does not preserve the predicate Rg, so

/0 1\ f01***--01-0 ll,

nionoioi-oi***--K0-

We see that in 2 x 12 matrix the elements in (l,i) and (1 ,j) positions coincide if and only if they coincide in (2,i) and (2,j) positions.

Thus, g ^ q ^ coincides with /1 ^ q ^ or coincides with /2 ^ q

It contradicts the fact that /1 and f2 preserve Rg.

The ESjj-closure of the remaining sets is verified similarly □

Theorem 2. For all Ki,..., Ku if i / j, then Ki <2 Kj.

Proof. The validity of the statement follows from Table 1. There is an unary multioperation / at the i-th row and the j-th column in the table such that f e Ki and f £ Kj. □

Table 1

The pairwise difference of sets Кi — Кц

Кг к2 Кз Ki Кь Кв к7 Ks К 9 к ю Ки

Ki X 0* 0* 00 01 00 00 00

к2 *1 X 11 — — *1 01 01 11 -- —

К3 ** 00 X 0- 0- ** 00 01 00 0- 00

** ** 11 X -1 ** 01 01 11 11 11

Кь ** ** 11 00 X ** 01 01 11 11 11

кв 11 00 11 00 -- X 00 11 00 -- —

к7 ** ** ** X 11 11 11 -1

к8 ** 00 ** 00 X 00 00 00

Kçi 10 10 10 10 -- ** 01 01 X — —

К10 ** ** 10 10 *— ** 01 01 — * X 10

Кгг ** ** +0 *— ** 01 01 — * 0111 X

3. Completeness criterion

Let fKi be a multioperation that does not belong to Щ (i € {1,..., 11}). Lemma 2. [0,1, /кб,/к6] = M2.

Proof. Obviously, [0,1, *, —] С [0,1, fx6, fK6}- In paper [9] it is proved that [0,1, *] =0\. In [1] it is shown that the set is precomplete in with respect to the composition operator. Thus, the lemma is proved. □

Lemma 3. If gi(x) = (—),g2(x) = (10), then[gl,g2, fKs, fKe, fKg] = M2.

Proof. Substitute gi(x) and g2{x) into the multioperation fxg. Consider the cases when we obtain multioperations u\(x) = (0—) and u2(x) = (—*). The other cases reduce to this one or Lemma 2.

Let v(xi,x2) = g2(x2,X2)- Substituting U2 and §2 in v we get

v{u2{x2), g2{x2)) = g2{u2 (x2), §2 (x2)) = (0*).

Using the multioperations g\ and (0*) it is easy to get a constant 0.

Substitute g2(x) into the multioperation fx6. We get four operations si (a;) = (0*), s2(x) = (1*), ss(x) = (—*), s^(x) = (**). Consider the last one. Using the equality predicate branching operator we get

./ n )92{xi), if X\ = x2] , , t(xi,x2) = < . =(1**0).

I 54(^1), otherwise;

Thus we have t((0—)(x2), (—0)(x2)) = (11). Now, by Lemma 2, we obtain that [gi,g2, ¡k5, fk&, ¡Kg] is complete in M2. □

Lemma 4. Ifg1(x) = (—),g2(x) = (11). Then [gi,g2, ¡k2, ¡k6, ¡k6, Jk7] is complete in M2.

Proof. It is enough to obtain the constant 0 and use Lemma 2.

The composition ¡k2( 1, •••,!) defines the unary operation v\ = (00) or v2 = (**). There is a binary set (a\,..., an) for the multioperation fx7 such that fK7(ot\,..., an) = 0.

Let h(x) = fK7(ui(x),... ,un(x)) and

fxi, if en = 0;

Ui(x) = <

if Cti = 1.

Then h(x) is one of the following operations:

ti = (00), t2 = (0*), i3 = (01), U = (0-).

Clearly, t2(g\(x)) = 0, therefore, it remains to consider two cases for ¿3 and ¿4. Let

p(x 1 x2) - iV2(Xl^ if Xl = X2]

' [¿3(^1) (or ¿4(^1)), otherwise.

The composition p(gi(x2),g2(x2)) defines the constant 0. □

Lemma 5. If gi(x) = (—),g2(x) = (1-), then [gi,g2, fx2, ¡k5, fx6, fx7] is complete in M2.

Proof. It is enough to obtain constant 0 or constant 1. One of 8 unary multioperations can be obtained by identifying variables in fx2 '■

hi = (00), h2 = (*0), hi = (0*), hA = (1*), h5 = (10), h6 = (-0), hj = (**), he = (-*).

The first four cases allow us to obtain the necessary constant.

Consider = (10). In this case, we have following operations g%{x) = (0—), Qa{x) = (—0), and g${x) = (—1). There is a binary set (a\,... ,an) for the multioperation fx6 such that fK6{ai, • • •, an) = *• Let us consider cases when a composition with an external operation fx6 and internal hs (or variables identification) defines unary operations t\(x) = (**) or t2{x) =

Using the equality predicate branching operator from multioperations (1010) and (****) we obtain u(x i,x2) = (1 * * 0). Superposition u(93(x2),54(^2)) defines the constant 1. To conclude the proof, it remains

Lemma 6. If gi{x) = (—),g2{x) = (00), then [gi,g2, fa2, fa6, fa6, fa9] is complete in

Proof. The proof of this statement is similar to the proof of Lemma 4. □

Lemma 7. If gi(x) = (—),g2(x) = (**), then \gi,g2, fa2, faB, fa6, fa7] is complete in

Proof. By identifying variables in fx7, we can obtain a multioperation h(xi,x2) such that on the binary sets (01) and (10) it takes one of the following four values: (00), (01), (0*), (0-).

Superposition t(x\, g\(x\)) defines unary multioperation p(x) = (0/5), where /3 € {0, *, 1, —}. In the first two cases the validity of the statement follows from Lemma 6. Consider p\(x) = (01) and p2(x) = (0—).

Using the equality predicate branching operator from multioperations pi{x2) and (****) we obtain (*10*). Superposition (*10*)(pi(a;i), <7i(a;i)) defines multioperation (10). Hence, the set [<71,g2, fx2, fx6, /к6> Îk7] is complete by Lemma 3.

Similarly, from p2 and * we get multioperation (* — 0 *). Superposition (* — 0 *)(xi,p2(xi)) defines multioperation (*0). Then we obtain (* 0 * 0)(p2(x2),gi(x2)) = (00). Finally, we have Lemma 6. □

Lemma 8. If gi(x) = (—),g2(x) = (*-), then \gi,g2, fa2, fa6, fae, fa7] is complete in M2.

Proof. Using the ideas and techniques from Lemma 7, we can obtain multioperation (0—). □

Theorem 3. A set of multioperations R is ESjj-complete if and only if it is not contained entirely in any of the classes K\ — Кц.

to note that h*,(t2(x2),gs(x2)) = (*0).

□

Let

t(x i,x2)

*, if x\ = x2; h(xi,x2), otherwise.

Proof. One of 8 unary multioperations can be obtained by identifying variables in /kh '■

fk n = (~), /in = (00), fKll = (11), /in = (10),

/in = (0-), /in = (-1), /in = (-0), /in = (I-)-

Since /in(/in(^)) = (—) and /in(/in(a:)) = (—), then it is enough to consider the first six operations.

Case 1. Consider fy = (—). By identifying variables in f^, we can obtain one of the following unary multioperations: (10), (11), (1*), (1—), (*0), (*1), (**), (*-).

It is clearly that it is enough to consider only six pairs of multioperations:

{(10), (—)} {(11), (--)} {(i-),(—)}

{(00), (—)} {(**),(—)} {(*-),(—)}•

However, we have Lemmas 3-8 for all these pairs.

Let us remark that (*1)(—)(x) = (11) and (*0)(—)(x) = (00).

Case 2. Consider = (00). Superposition fx3 and defines multioperation (11) or (—).

It follows from Lemma 2 that the set {(00), (11)} is complete in M2. For the set of multioperations {(00), (—)} we use Lemma 6.

Case 3. Consider = (11). Superposition fx4 and f^ define

multioperation (00) or (—).

For sets {(00), (11)} and {(11), (—)} we use Lemma 2 and Lemma 4 respectively .

Case 4- Consider = (10). There are a pair of binary sets (a) and (a) for the multioperation fx10 such that (/'k10(&)/k10(&)) G {(00), (11), (0-), (-0), (1-), (-1), (11)}.

Substitute (10) into the multioperation fx10 ■ We obtain one of the following unary multioperations: (00), (11), (0—), (—0), (1—), (—1), (--)• Using

ideas from cases 1-3 for these multioperations, we complete the proof.

Case 5. Consider fKii = (0-). Note that fKs(0,..., 0) € {1, -}. Therefore we have /K3(/in) € {(11), (1—), (—)}• As above, we use ideas from cases 1-3.

Case 6. Consider fbKii = (-1). We have fKi{ 1,..., 1) € {0,-}. Then we can obtain /K4(/in) € {(00), (—0), (—)} and use cases 1-3.

This completes the proof of the main theorem. □

4. ESu-closed subsets of Kg

In this section, we show that precomplete set Kg consists of 20 ESu-closed subsets.

Lemma 9. Any ESu-closed class from M2 is generated by the set of all its multioperations depending on at most two variables.

Proof. This Lemma can be proved by methods of the corresponding statement from [9]. □

According to the previous Lemma, it can be easily checked that Kg contains the following 16 multioperations only:

Kg = {(****),(*--*), (*01*), (*10*), (— * * —), (----),

(-01-), (-10-), (0 * *1), (0 - -1), (0011), (0101), (1 * *0), (1 - -0), (1010), (1100)} Lemma 10. [(* * * *), (1--0)] = Kg.

Proof. To prove Lemma, we obtain the remaining 14 multioperations using 9i(x,y) = (****) and g2(x,y) = (1 - -0). We get

(----) =92(х,д2(х,у)У, (10) = g2(x,x); (01) = g2(g2(x, x)).

All other multioperations in Kg are obtained from the following consideration. Let /1 = (a\a2asa4) and f2 = (/З1/З2/З3/З4). If /1, /2 € Kg, then we can obtain /3 = (а^/Зг/Ззщ) and /4 = (0.10.3^0.4), where /3, /4 € Kg. □

Let Pi = {/ I /(0,..., 0) <E {*} and P2 = {/ | /(0,..., 0) € {-}}. It is easy to prove that P\ and P2 are ESu-closed sets. Now consider the following 20 subsets of Kg.

Si = {(* * **)}; S2 = {(----)}; S3 = {(* * **), (*--*)};

= {(0011), (0101)}; = {(-**-),(----)};

S"6 = {(* * **), (*01*), (*10*)} ; 5*7 = {(0**1), (0011), (0101)};

Ss = {(0--1), (0011), (0101)}; = {(----), (—01—), (—10—)};

Sw = {(0011), (0101), (1010), (1100)} ;

S11 = {(* * **), (*--*), (- * *-), (----)};

¿>12 = {(* * **), (*--*), (*01*), (*10*)};

S13 = {(0**1),(0--1), (0011), (0101)}; Su = {("**"),(----),(-01-),(-10-)};

Si5 = {(* * **),(*01*),(*10*),(0 * *1),(0011), (0101)};

Sie = {(----), (-01-), (-10-), (0 - -1), (0011), (0101)};

Sn = {(* * **), (* - -*), (*01*), (*10*), (0 * *1), (0 - -1), (0011), (0101)};

Sis = {(- * *-), (----), (-01-), (-10-), (0 * *1), (0 - -1),

(0011),(0101)};

519 = {(* * **), (*01*), (*10*), (0 * *1), (0011), (0101), (1 * *0), (1010), (1100)};

520 = {(----), (-01-), (-10-), (0 - -1), (0011), (0101),

(1--0), (1010), (1100)};

Theorem 4. The sets S\ — S20 a/re ESu-closed.

Proof. It is evident that Si, S2, s3, S4, and S5 are ESu--closed. Matveev in [11] proved the ESu-closure of S&, s7, S10, ¿>15, and S19. We define the remaining sets as intersections of known ESu-closed classes.

Ss = KinK3nK6nKg-, Sg = P2 n Kq n Kg-,

Sn=K7nKg-, SU=PiriKg-,

S13 = Ki n K3 n Kg] Su = P2 fl Kg]

Sm =KinK6nKg] Su = K3nKg]

Sis=Kif]Kg] S20=K6nKg.

It proves that all Si — S20 are ESu--closed. □

Remark 1. Closed sets from [11] can be represented as follows.

S6 = PinK3nK5nKg] S7 = KinK3nK5nKg] Sio = K5 n K& n Kg] <§15 = K3 n K§ Pi Kg]

siq = K5nKg.

Lemma 11. Let Q be an any ESu-closed subset of Kg. Let g(x) be an unary multioperation obtained by identifying variables of some

f{x 1, ...,xn)eQ.

Then g(x) can be one of the following three multioperations:

g{x) = (01), g{x) = (**), g{x) = (—).

Proof. Consider f(xi,...,xn) € Q. Since Q C Kg, it follows that / € Pol Rg. Therefore,

f(x,...,x) € {(**),(—), (01), (10)}.

However (01)(x) = (10)(10)(a:). □

Corollary 1. There are three minimal subsets of Kg only.

In the following lemmas, we will keep in mind that the sets under consideration are precomplete only in given sets.

Lemma 12. 6*11 is an E Su -precomplete set in Q iff [Q\ = Kg.

Proof. Since (* * **) € 5*11, we need to obtain the multioperation (1--0).

We consider various cases for /sn.

Let fsu = (ICK1CK2O), where 0:1,0:2 G 0,1, —}. Hence,

(1__0) = [fSl1 [ix = y>

1(----){x,y), otherwise.

For the set of multioperations {(* * **), (1--0)} we use Lemma 10.

Let /зц = (Oo^l), where 0:1,0:2 € {*,0,1, —}. Now we obtain

(*01*) = {(****)(ж'у)' [ix = y] 1 fsu(x,x), otherwise;

and

O-P1P2O) = (*01*)((----)(x,y),fSll(x,x)),

where [3\, (З2 € {*, 0,1, —}. Then we use the previous case. Let fsn = (01OI02), where 01,02 € {*, —}. We obtain

\fs

u(.x,y), otherwise;

and

(1 * *0) = (*01*)((— * *—)(x,y),x).

Then we use the previous case.

Finally, let Jsu = (011002), where 01,02 € {*, —}. We can obtain

(oi01o2) = fsuiVyx).

This completes the proof. □

Lemma 13. Si7 is an ESu-precomplete set in Q iff [Q\ = Kg.

Proof. Since (* * **) € S17, we need to obtain the multioperation (1--0).

Let fs17 = (I01O2O), where Oi, 02 € {*, 0,1, —}. Then

(1__0) = [ix = y;

I (*--*)(x,y), otherwise.

Now we get i?S{/-complete set in Kg.

Let fs17 = (—0i02—), where 01,02 € {*,0,1,—}. Using the operator with the equality predicate branching, we get (—**—) and then

(1 * *0) = (*01*)((— * *—)(x,y),x).

Finally, we use ideas from Lemma 12. □

Lemma 14. Sis is an E Su -precomplete set in Q iff [Q\ = Kg.

Proof. Let fs18 = (*0i02*), where oi, 02 € {*, 0,1, —}. The multioperation (**) can be obtained by identifying variables in fs18. Using the operator with the equality predicate branching, we obtain (*01*) and then we use ideas from the final part of Lemma 12.

Let fs18 = (I01O2O), where oi, 02 € {*, 0,1, —}. Then we obtain

0/?l/?2*) = (0**1) (x, fs18(x, y)) and use the previous case. □

Lemma 15. S19 is an ESu-precomplete set in Q iff [Q] = Kg.

Proof. The multioperation (—) can be obtained by identifying variables in fSig. Using (1 * * 0), we get

Lemma 17. ¿>12 is an E Su -precomplete set in Q iff [Q] = ¿>17.

Proof. Note that ¿>12 contains four multioperations from ¿>17. Let fs12 = (Ocki^I), where ai,a2 € {*, 0,1, —}. Using the operator with the equality predicate branching, we can get the remaining four multioperations.

If fs12 = (—aia2—), then using _E-operator, we obtain the whole set of 5*11 and some other operations. By Lemma 12, we get Kg.

If fs12 = (ICK1CK2O), then using _E-operator, we obtain the whole set of ¿>19 and some other operations. So, by Lemma 15, we get Kg. □

Lemma 18. Si5 is an ESu -precomplete set in Q iff [Q\ = S17 or

Proof. Consider cases for fs16.

Case 1. Let fs15 £ K3 and fs15 ^ K5. If fs15 = (1--0), then we have a

complete set for Kg. If fs15 = (—aia2—), where 0:1,0:2 € {*,0,1, —}, then using _E-operator, we obtain the whole set of Su. Therefore we get Kg.

Case 2. Let fs15 € K3 and fs15 £ If fs15 = (0--1), then we obtain

(*--*). It gives us S17. Similarly, for fs15 = (*--*) we obtain S17.

Case 3. Let fs15 £ K3 and fs15 € K5. Let fs15 = (lo^O), where oi, 02 € {*, 0,1}. Using _E-operator, we obtain the whole set of S19. □

Lemma 19. S13 is an ESu -precomplete set in Q iff [Q\ = S17 or

Proof. Case 1. Let fs13 £ K\ and fs13 £ K3. Let fs13 = (101020), where oi, 02 € {*, 0,1, —}, then we obtain

□

Lemma 16. S20 is an E Su -precomplete set in Q iff [Q] = Kg. Proof. The proof is trivial.

□

[Q] = S19.

[Q] = S is-

(*PlP2*) = (fi**l)(x,fSl3(,x,y))

and

(1 - -0) =

fs13(x,y), if X = y] (0--1 )(x,y), otherwise;

Therefore we get Kg.

Case 2. Let fs13 € K\ and fs13 £ K3. Let fs13 = (—0102—), where 0:1,0:2 € {*,0,1,—}. Using _E-operator, we can get the remaining four multioperations for Sn.

Case 3. Let fs13 £ K\ and fs13 € K3. Let fs13 = (*0i02*), where 01,02 € {*,0,1,—}. Using _E-operator, we can get the remaining four multioperations for Sis- □

Lemma 20. ¿>14 is an ESu-precomplete set in Q iff [Q\ = Sis-

Proof Case 1. Let fs14 £ P2 and fs14 € K\. Let fs14 = (0aia2l), where 01,02 € {*,0,1,—}. Using _E-operator, we can get the remaining four multioperations for S\s-

Case 2. Let fs14 £ K\. If fs14 = (*0i02*), then using _E-operator we obtain the whole set of Sn- Adding (—01—) to Sn, we get Kg. If fs14 = (I01O2O), where oi, 02 € {*, 0,1, —}, then we obtain

(*/?l/?2*) = (- * *-) {x,fs14(x,y))

and

(1--0) = J-^14^'^' [ix = y;

1 (----)(x,y), otherwise.

Therefore we get Kg. □

Lemma 21. S\e is an ESu -precomplete set in Q iff [Q\ = S\& or

[Q] = S20.

Proof. Consider the various cases for fs16.

Case 1. Let fs16 £ K\ and fs16 £ K§. If fs16 = (1**0), then we obtain

(****) = fs16{x,fs16(x,y))

and

(1--0) =

fsle(x,y), if x = y]

(0--1 )(x,y), otherwise.

Therefore we get Kg.

If fs16 = (*0i02*), where 01,02 € {*,0,1}. The multioperation (**) can be obtained by identifying variables in fs16. Using (0101), we get

(*io*) = [ix=y>

1 (0101) (a;, y), otherwise;

and

(1 0) = (*10*)((0 l)(x,y), (----)(x,y)).

Now we get Kg.

Case 2. Let fs16 € K\ and fs16 £ K§. Let fs16 = (0**1), then we obtain (— * *_). it gives us ¿>i8.

Case 3. Let fs16 £ K\ and fs16 € Kë. Let fs13 = (laia^O), where ai,a.2 € {0,1, —}. Using _E-operator, we can get the remaining three mul-tioperations for ¿>20- □

Proposition 1. [(1 * *0)] = ¿>19.

Proposition 2. [(1 - -0)] = S20.

Propositions 1 and 2 were proved in [11] and [14] respectively.

Lemma 22. S10 is an ESu-precomplete set in Q iff [Q\ = ¿>19 or

[Q] = S20.

Proof Consider the various cases for fs10.

Case 1. Let fs10 £ K$ and fs10 £ K§. If fs10 = (—**—), then we obtain

(****) = fSw (x, (1100)(a;, y)).

Moreover,

(-10-) = [ix = y;

1 (1100) (a?, 2/), otherwise;

and

(1 0) = (—10—) {x, (1010)(a;, y)).

It follows that we get Kg.

If fs10 = (*--*)• The multioperation (**) can be obtained by identifying variables in fs10. Using (1010), we get

(1 _ _ 0) = ji1010)^)' if x = V'

\fs10(x,y), otherwise.

Now we get Kg.

Case 2. Let fSl0 € K5 and fSl0 £ Ke. If fSl0 = (*aia2*), where ol\,ol2 € {*,0,1}. The multioperation (**) can be obtained by identifying variables in fs10. Moreover,

(1^Q)= i(1010)(a;,y), ifx = y, 1 (**)(a?), otherwise.

Thus we have ^¿>{/-complete set for S19 (see Proposition 1). Trivially, if fSl0 = (0 * *1) or fSl0 = (1 * *0) we get S"i9. Case 3. Let fs10 £ K$ and fs10 € K&. This case is similar to the previous one. Using Proposition 2, we get ¿>20- □

82 V. I. РАМТЕЬЕЕУ, Ь. V. ШАВЕТЭ

Ьетта 23. ТИе /оНомтпд з1а1етеШз аге 1гие.

1) ¿>1 гз ап Еви-ргесотрЫе гп С} ^ [0\ = 5з ог [0\ = 5б;

2) ¿>2 is ап Еви-ргесотрЫе гп С} ^ [0\ = ¿>5 ог [0\ = ¿>9;

3) 5*3 гз ап Еви-ргесотрЫе гп <5 Ш [0\ = 5*11 ог [0\ = 5*12;

4) Б4 ¿5 ап Еви-ргесотрЫе гп <5 Ш [0\ = 5*7, ог [ф] = 5в, ог

И = 5ю;

5) 5б г« ап Еви-ргесотрЫе гп <5 Ш [0\ = 5*11 ог [0\ = 5*14;

6) 5б гв ап Еви-ргесотрЫе гп <5 г^ [ф] = 512 ог [ф] = 515; 7,) 5г г« ап Еви -ргесотрЫе гп <5 Ш [0\ = 5*13 ог [ф] = 515; 8) 5в г« ап Еви-ргесотрЫе гп <5 Ш [0\ = 5*13 ог [0\ = 516; ^ 5э гз ап Еви-ргесотрЫе гп <5 Ш [0\ = 5*14 ог [ф] = 516-

ТЬе ргоо£ ¡в огшШё. II ивев Ше 1ёеав апё 1ес]ашс1ие8 &от ргеуюив Ьеттав.

ТЪеогет 5. ТИе Еви-ргесотрЫе Кд сотгзЬз о/ 20 Еви-сЬвес!, зибзе^.

1п сопсЫвюп, \тс ргевегй Ше st.ruct.ure с^ Кд т dia.gra.rn 1.

Р1д-иге 1. ЕЯи-с1ове(1 subsets Кд

5. Conclusion

In this paper, we considered precomplete ESj-closed classes of multi-operations defined on a 2-element set. The next steps are to obtain all ESj-closed classes of M2 and to determine properties of multioperations defined on a 3-element set.

References

1. Doroslovacki R., Pantovic J., Vojvodic G. One Interval in the Lattice of Partial Hyperclones. Chechoslovak Mathematical Journal, 2005, no. 55(3), pp. 719-724. https://doi.org/10.1007/s10587-005-0059-0.

2. Lau D. Function Algebras on Finite Sets. A basic course on many-valued logic and clone theory. Berlin, Springer-Verlag, 2006, 668 p.

3. Lo Czu Kai. Maximal closed classes on the Set of Partial Many-valued Logic Functions. Kiberneticheskiy Sbornik, Moscow, Mir Publ., 1988, vol. 25, pp. 131-141. (in Russian)

4. Lo Czu Kai. Completeness theory on Partial Many-valued Logic Functions. Kiberneticheskiy Sbornik, Moscow, Mir Publ., 1988, vol. 25, pp. 142-157. (in Russian)

5. Machida H. Hyperclones on a Two-Element Set. Multiple-Valued, Logic. An International Journal, 2002, no. 8(4), pp. 495-501. https://doi.org/10.1080/10236620215294.

6. Machida H., Pantovic J. On Maximal Hyperclones on {0,1} — a new approach. Proceedings of 38th IEEE International Symposium on Multiple-Valued, Logic (ISMVL 2008), 2008, pp. 32-37.

7. Marchenkov S.S. On the Expressibility of Functions of Many-Valued Logic in Some Logical-Functional Classes. Discrete Math. Appl., 1999, vol. 9, no. 6, pp. 563-581.

8. Marchenkov S.S. Closure Operators with Predicate Branching. Bulletin of Moscow State University. Series 1. Mathematics and Mechanics, 2003, no. 6, pp. 37-39. (in Russian)

9. Marchenkov S.S. The Closure Operator with the Equality Predicate Branching on the Set of Partial Boolean Functions. Discrete Math. Appl., 2008, vol. 18, no. 4, pp. 381-389. https://doi.org/10.1515/DMA.2008.028.

10. Marchenkov S.S. The E-closure Operator on the Set of Partial Many-Valued Logic Functions. Mathematical problems in cybernetics, Moscow, Fizmatlit, 2013, vol. 19, pp. 227-238. (in Russian)

11. Matveev S.S. Construction of All E-closed Classes of Partial Boolean Functions. Mathematical problems in cybernetics, Moscow, Fizmatlit Publ., 2013, vol. 18, pp. 239-244. (in Russian)

12. Panteleev V.I. The Completeness Criterion for Depredating Boolean Functions. Vestnik of Samara State University. Natural Science Series., 2009, no. 2 (68), pp. 60-79. (in Russian)

13. Panteleev V.I. Completeness Criterion for Underdetermined Partial Boolean Functions. Vestnik Novosibirsk State University. Series Mathematics, 2009, vol. 9, no. 3, pp. 95-114. (in Russian)

14. Panteleev V.I., Riabets L.V. The Closure Operator with the Equality Predicate Branching on the Set of Hyperfunctions on Two-Element Set. The Bulletin of Irkutsk State University. Series Mathematics, 2014, vol. 10, pp. 93-105. (in Russian)

15. Romov B.A. Hyperclones on a Finite Set. Multiple-Valued, Logic. An International Journal, 1998, vol. 3(2), pp. 285-300.

Vladimir Panteleev, Doctor of Sciences (Physics and Mathematics), Professor, Institute of Mathematics, Economics and Informatics, Irkutsk State University, Russian Federation, 664000, Irkutsk, K. Marx st., 1, tel.:+7(3952)242214 (e-mail: [email protected])

Leonid Riabets, Candidate of Sciences (Physics and Mathematics), Associate Professor, Institute of Mathematics, Economics and Informatics, Irkutsk State University, Russian Federation, 664000, Irkutsk, K. Marx st., 1, tel.:+7(3952)242214 (e-mail: [email protected])

Received 05.08.19

Критерий полноты для оператора замыкания с разветвлением по предикату равенства на множестве мультиопе-раций ранга 2

В. И. Пантелеев, Л. В. Рябец

Аннотация. Мультиоперадии представляют отображения, задаваемые на конечном множестве и возвращающие в качестве своих значений все подмножества рассматриваемого множества. Оператор суперпозиции приводит к континууму замкнутых множеств. Поэтому возникает необходимость рассмотрения операторов замыкания, которые наряду с суперпозицией содержат другие операции. В работе рассматривается замыкание мультиопераций, полученное применением оператора суперпозиции, основанной на объединении, оператора разветвления по предикату равенства. Для мультиопераций, задаваемых на двухэлементном множестве, указаны все предполные множества, сформулирован и доказан критерий полноты. Приведена диаграмма включений замкнутых классов для одного из предполных классов.

Ключевые слова: замыкание, предикат равенства, мультиоперация, замкнутое множество, суперпозиция, критерий полноты.

Список литературы

1. Doroslovacki R., Pantovic J., Vojvodic G. One Interval in the Lattice of Partial Hyperclone // Chechoslovak Mathematical Journal. 2005. N 55(3). P. 719-724. https://doi.org/10.1007/sl0587-005-0059-0

2. Lau D. Function Algebras on Finite Sets. A basic course on many-valued logic and clone theory. Berlin : Springer-Verlag, 2006. 668 p.

3. Jlo Джукай. Максимальные замкнутые классы в множестве частичных функций многозначной логики // Кибернетический сборник. Новая серия. М. : Мир, 1988. Вып. 25. С. 131-141.

4. Jlo Джукай. Теория полноты для частичных функций многозначной логики // Кибернетический сборник. Новая серия. М. : Мир, 1988. Вып. 25. С. 142-157.

5. Machida H. Hyperclones on a Two-Element Set // Multiple-Valued Logic. An International Journal. 2002. N 8(4). P. 495-501. https://doi.org/10.1080/10236620215294

6. Machida H., Pantovic J. On Maximal Hyperclones on {0,1} — a new approach // Proceedings of 38th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2008). 2008. P. 32-37.

7. Марченков С. С. О выразимости функций многозначной логики в некоторых логико-функциональных языках // Дискретная математика. 1999. Т. 11, вып. 4. С. 110-126. https://doi.org/10.4213/dm400

8. Марченков С. С. Операторы замыкания с разветвлением по предикату // Вестник МГУ. Сер. 1, Математика и механика. 2003. № 6. С. 37-39.

9. Марченков С. С. Оператор замыкания с разветвлением по предикату равенства на множестве частичных булевых функций // Дискретная математика. 2008. Т. 20, вып. 3. С. 80-88. https://doi.org/10.4213/dml015

10. Марченков С. С. Оператор Я-замыкания на множестве частичных функций многозначной логики // Математические вопросы кибернетики. М. : Физмат-лит, 2013. Вып. 19. С. 227-238.

11. Матвеев С. А. Построение всех Я-замкнутых классов частичных булевых функций // Математические вопросы кибернетики. М. : Физматлит, 2013. Вып. 18. С. 239-244.

12. Пантелеев В. И. Критерий полноты для доопределяемых булевых функций // Вестник Самарского государственного университета. Естественнонаучная серия. 2009. № 2 (68). С.60-79.

13. Пантелеев В. И. Критерий полноты для недоопределенных частичных булевых функций // Вестник НГУ. Сер. Математика, механика, информатика. 2009. Т. 9, вып. 3. С. 95-114.

14. Пантелеев В. П., Рябец J1. В. Оператор замыкания с разветвлением по предикату равенства на множестве гиперфункций ранга 2 // Известия Иркутского государственного университета. Сер. Математика. 2014. Т. 10. С. 93-105.

15. Romov В. A. Hyperclones on a Finite Set // Multiple-Valued Logic. An International Journal. 1998. Vol. 3(2). P. 285-300.

Владимир Иннокентьевич Пантелеев, доктор физико-математических наук, профессор, Иркутский государственный университет, Российская Федерация, 664000, Иркутск, ул. К. Маркса, 1 тел.: +7(3952)242214 (e-mail: [email protected])

Леонид Владимирович Рябец, кандидат физико-математических наук, доцент, Иркутский государственный университет, Российская Федерация, 664000, Иркутск, ул. К. Маркса, 1 тел.: +7(3952)242214 (e-mail: [email protected])

Поступила в редакцию 05.08.19